# Re: subrings of ZxZ

*From*: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)*Date*: Sun, 12 Oct 2008 02:53:45 +0000 (UTC)

In article <518cbdf5-ae9c-4590-a023-eec81e4a9ded@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,

Luke Wu <LookSkywalker@xxxxxxxxx> wrote:

What are all the subrings of ZxZ if we require (0,0) and (1,1) to be

in the subring?

the whole ring is a subring

subset of all pairs of form (a,a) is a subring

I believe that's all. Am I right?

No. For example, what about the collection of all pairs (a,b) in which

a-b is a multiple of 5?

This contains (0,0) and (1,1), and (a,a) for all a. If b=a+5k and

d=c+5m, then (a,b) + (c,d) = (a,a+5k) + (c,c+5m) = (a+c,(a+c)+5(k+m)),

so the set is closed under sums. It is also closed under differences;

what about products?

--

======================================================================

"It's not denial. I'm just very selective about

what I accept as reality."

--- Calvin ("Calvin and Hobbes" by Bill Watterson)

======================================================================

Arturo Magidin

magidin-at-member-ams-org

.

**References**:**subrings of ZxZ***From:*Luke Wu

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